In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form ∑ i = 1 ∞ r i x i {\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}} where x 1, x 2, … {\displaystyle x_{1},x_{2},\ldots } are all elements of a topological vector space X {\displaystyle X}, and all r 1, r 2, … {\displaystyle r_{1},r_{2},\ldots } are non-negative real numbers that sum to 1 {\displaystyle 1}.
